Questions tagged [line-bundles]

For questions about line bundles, that is vector bundles of rank $1$, over topological spaces.

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dimension of linear system and multiplicity at a point

Prove the following statement. Let $X$ be a smooth projective surface and let $L$ be a line bundle on $X$. For $x\in X$ if $h^0(|L|)\geq\frac{m(m+1)}{2}$ then $|L|$ contains a curve $C$ passing ...
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31 views

Existence of a global section of $L(D)$ vanishing on $D$.

Given a complex surface $X$ and $D$ a smooth curve on $X$, there is an associated line bundle $L(D)$ over $X$. Is there always a global section $s\in H^0(X,L(X))$, s.t. $s$ is just vanishing on $D$?
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52 views

Metric on the dual line bundle

Let $X$ be a compact Kähler manifold and $L$ be a holomorphic line bundle on $X$ with a Hermitian metric $h$. I am struggling to understand how one induces a canonical dual metric $h^*$ on $L^*$. Now ...
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1answer
51 views

On vanishing of global sections of some line bundles

Let $X$ be a smooth projective surface $S$ in $\mathbb P^3$. Let $C$ be a smooth hyperplane section of $S$ and $D$ be a non-zero divisor on $S$. Consider the short exact sequence : $0 \to \mathcal O_S(...
2
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1answer
47 views

Pull-back of the diagonal is the negative canonical bundle

Let $\Delta : X \rightarrow X \times X$ denote the diagonal embedding where $X$ is a $\mathbb{C}$-variety. In a paper I am reading it asserts that the pull-back of $X$ diagonally embedded in $X \times ...
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1answer
20 views

Definition of index of a line bundle section is well-defined?

I'm reading lecture notes on advanced complex analysis and I'm struggling with a certain claim about the definition of the index of a complex line bundle section. Let $L\to M$ be a complex line bundle ...
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1answer
38 views

Complex line bundles over a Riemann surface can be given a holomorphic line bundle structure

In p.20 of this lecture note (link: http://www.math.ubc.ca/~cautis/math428/notes-bundles.pdf), it is written that every complex line bundle over a Riemann surface can be given a holomorphic line ...
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64 views

Number of sections of line bundles on a plane curve

Let $A$ be a line bundle of degree $1$ on a smooth plane quintic curve over complex numbers. We know that it can have at most $1$ section. My question is the following : Under what conditions it has ...
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30 views

Connection 1-form, symplectic potential?

I recently encountered a formula that fell a little bit from the sky: Given: A symplectic manifold $(M,\omega)$, a Hermitian line bundle $\pi:B\rightarrow M$, a connection $\nabla$ on $B$ and a ...
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83 views

How can I use Grothendieck--Riemann--Roch theorem?

For my research, I'm trying to understand how the Grothendieck--Riemann--Roch theorem is used in the paper The Birational Geometry of the Hilbert Scheme of Points on Surfaces by Aaron Bertram & ...
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1answer
50 views

Projective bundle formula for sheaf cohomology

Given a projective bundle $p:P(E)\rightarrow X$ associated to a vector bundle $E$ and a line bundle on $P(E)$ of the form $\mathcal{O}_{P(E)}(1)\otimes p^*L$ where $L$ is a line bundle on $X$, is ...
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1answer
38 views

Is it possible for the pullback of an ample line bundle under projection to be big?

Given an ample line bundle on a curve is there any chance for its pullback to the projective bundle of some vector bundle to be big? It is known that first cohomology of inverse of nef and big line ...
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1answer
39 views

Pullback of hermitian line bundle

let $f:X\to Y$ be a holomorphic map between complex manifolds. Let $(L,h)$ be a hermitian line bundle on $Y$. Namely $L$ is a holomorphic line bundle on $Y$ and $h$ is a $C^{\infty}$-hermitian inner ...
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1answer
51 views

Line bundles on $S^2$ and $\pi_2(\mathbb{R}P^2)$

Real line bundles on $S^2$ are all trivial, but what about the following way to think about a line bundle: we view a line bundle on $S^2$ (thought of as living in $3$d space) as providing a map $S^2 \...
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1answer
38 views

Global sections of powers of a line bundle which is of the form $L' \otimes L$, where $L'$ is globally generated and $L$ is a torsion element.

Let $Y$ be an irreducible complex projective variety of $\mathrm{dim}(Y) \geq 1$ and $X$ an irreducible complex projective variety. Assume that $M \in \mathrm{Pic}(Y)$ is a torsion element, say of ...
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50 views

Fano surface vs. del Pezzo surface

This article from wikipedia defines a Fano variety as a complete variety whose anticanonical bundle is ample. It also states that: A Fano surface is also called a del Pezzo surface. Every del Pezzo ...
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32 views

Noether folrmula, Riemann Roch , Reimann Surfaces, Global holomorphic sections of a Line bundle

Let $K$ be the canonical line bundle of a Riemann Surface $M$ of genus $g.$ Consider the pullback of $K$ on $M \times M$ via projection on the first factor. What is the dimension of the global ...
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37 views

Curvature form on Riemann Surfaces

I am trying to understand the basic construction in P. Biran's paper "Lagrangian Barriers and Symplectic Embeddings". At the begining (2.1), there is a construction which relies on the ...
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67 views

Holomorphic line bundles with trivial Chern class are flat

Let $X$ be a complex, projective algebraic variety and let's work in the differential-complex setting. Let $L$ be a non-trivial hermitian holomorphic line bundle and assume that $c_1(L)=0$. Can we ...
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1answer
39 views

Construction of a quotient line bundle of a semistable vector bundle

I want to prove the following Lemma, but I'm stuck. Let $X$ be a smooth projective curve over $\mathbb{C}$ of genus $g\geq3$ and $E$ be a semistable vector bundle of rank $n$ and degree $d$ with $0\...
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40 views

Application of gonality of a plane curve

Let $X$ be a smooth plane projective curve (over $ \mathbb C$) of degree $5$ and genus $6$. Then we know that it has gonality $4$. This means the minimum degree of a base point free line bundle with ...
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1answer
59 views

How to see this identification of a Springer fiber with the bundle $\mathcal{O}_{\mathbb{P}}(2)$

I'm reading some notes and I come across this computation in the context of computing Springer fibers: Let $V$ be a $4$ dimensional vector space over $k$ with standard basis $e_i$. We have a nilpotent ...
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1answer
62 views

Does the restriction of $\mathcal{O}(1)$ to a quadric have a square root?

Let $Q^n$ be a smooth $n$-dimensional quadric in $P^{n+1}(\mathbb{C})$. Does the restriction of $\mathcal{O}(1)$ to $Q^n$ have a square root, for any $n \geq 1$? If $n = 1$, then $Q^1$ is ...
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54 views

When a line bundle on a projective variety separates tangent vetor?

In the Huybrecht's book Fourier Mukai transform in Algebraic geometry, in order to prove Bondal and Orlov's results (proposition 4.11), it seems that he uses following things. Suppose $k$ is an ...
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49 views

Given an element $\omega \in H^{(1,1)}(X,\mathbb{C})$ how to construct a line bundle with chern class $\omega$?

Let $X$ be a Kahler complex variety of dimension $1$. Let $\omega\in H^{(1,1)}_{dR}(X,\mathbb{C})$ such that $\int_{X}\omega = 0$. Can we find a line bundle $L$ over $X$ such that $c_1(L) = \omega$? ...
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1answer
116 views

How should I understand Kodaira dimension?

Say we have projective variety $X$. Its Kodaira dimension $\kappa(X)$ is defined by the “growth exponential” of $P_d := \dim H^0(X,K_X^{\otimes d})$ with respect to $d$, i.e. $\kappa(X) := -\infty$ ...
4
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1answer
110 views

Geometric meaning of the degree of the normal bundle $\mathcal{N}_{C/X}$

Assume all varieties projective and smooth over $\Bbb{C}$. Let $X\subset\Bbb{P}^3$ be surface and $C\subset X$ a curve in it. The normal bundle $\mathcal{N}_{C/X}$ is the cokernel of the map $T_C\...
2
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1answer
154 views

Divisor of a rational section in Ravi Vakil's notes

The following is from Exercise 14.2.A of Ravi Vakil's algebraic geometry notes (page 401 here). The exercise asks us to consider the rational section $\frac{x^2}{x+y}$ of the sheaf $\mathcal{O}(1)$ on ...
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1answer
69 views

Ampleness of a line bundle on the generic fibre

Let $f:X\rightarrow S$ be a morphism of regular schemes and $\mathscr{L}$ a line bundle over $X$. Suppose that $S$ is irreducible and let $\eta$ be its generic point. Assume that $\mathscr{L}\otimes\...
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73 views

Map to Proj of ring of sections

Let $X$ be an $A$-scheme ($A$ a ring) and $\mathscr{L}$ be a line bundle on $X$. Define the ring of sections of $\mathscr{L}$ to be $R(X,\mathscr{L})=\bigoplus _{n\geq 0} \Gamma(X,\mathscr{L^{\otimes ...
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55 views

Weil divisor arising from invertible sheaf and rational section?

I know there have been questions about this but I'm afraid I am still not really getting it. It's kind of embarrassing that something is not clicking at this point because I have asked some related ...
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50 views

Canonical divisor and canonical bundle

Let $X$ be a compact Riemann surface and the canonical divisor $K_X:=(\Omega)$, where $\Omega$ is a meromorphic top form on $X$. Question 1: Can I show the canonical divisor is the corresponding ...
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1answer
38 views

Valuation of a rational section of an invertible sheaf at a codim 1 point

Let $X$ be an integral scheme. Let $\mathcal{L}$ be an invertible sheaf. By a rational section of $\mathcal{L}$ I mean a pair $(s, U)$ where $s$ is a section of $\mathcal{L}$ over $U$ defined up to ...
4
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1answer
54 views

On a variety, every line bundle $L$ is isomorphic to $\mathcal{O}(D)$ for some Cartier divisor $D$?

I'm trying to read a part of the proof of Lemma 2.2 in Fulton's Intersection theory. He wants to prove that every line bundle $L$ on a variety (integral, of finite type over $k$) $X$ is of the form $\...
4
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1answer
126 views

Relationship between the ideal of an effective Cartier divisor and its invertible sheaf

Let $X$ be a scheme. We'll assume it's noetherian to avoid any pathologies. Let $D$ be an effective Cartier divisor on $X$. I am having trouble understanding how to go between the language of ...
3
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1answer
78 views

Explicit description of $\mathcal{O}_{\Bbb{P}^1}(-1)$ as a line bundle

I understand the construction of $\mathcal{O}_{\Bbb{P}^1}(-1)$ as a sheaf on $\Bbb{P}_\Bbb{C}^1$, but I'm trying to understand how exactly does this define a line bundle and why people call this the &...
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31 views

What is the connection between divisors and line bundles? [duplicate]

I'm somewhat new to algebraic geometry. I'm currently studying algebraic curves primarily over closed fields, (for future discussion let's just call such a curve C). I was taught to think of things ...
2
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1answer
104 views

Relation between tautological line bundle and blow up at the origin

We can define the projective $n$-space $\mathbb{P}^n$ as the quotient of $\mathbb{C}^{n+1}\setminus \{0\}$ by the action of $\mathbb{C}^*$ with all weights equal to $1$. Moreover we can define the ...
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33 views

Local class group and Class group of localizations

Let $R$ be a Noetherian normal domain with divisor class group $Cl(R)$ and Picard group $Pic(R)$ and we can consider $Pic(R)$ as a subgroup of $Cl(R)$. Consider the following two statements: (1) There ...
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1answer
92 views

When does the Picard group embeds inside the divisor class group?

Let $(X, \mathcal O_X)$ be a Noetherian, separated, integral scheme that is locally regular in codimension $1$ (i.e. if $\dim \mathcal O_{X,x}=1$ then $\mathcal O_{X,x}$ is regular). Then, is it true ...
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1answer
108 views

Prove is a line bundle

I want to prove that a specific object is a line bundle. Consider a normal variety $X$ and let $E$ be a line bundle on $X$. Denote by $s:X\to E$ the zero section, and consider $$F=(E\setminus s(X))\...
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15 views

holomorphic section of positive line bundle

I read the following statement from the book "L^2 approaches in several complex variables" page 206: Positive dimensional analytic sets must intersect with the zeros of holomorphic sections of ...
2
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1answer
125 views

Identify a pullback line bundle on $\mathbb{P}^1$

Consider a degree $d$ map $f:\mathbb{P}^1 \to \mathbb{P}^m$ for $d \geq 1$ and $m \geq 2$, together with a line bundle $\mathcal{O}_{\mathbb{P}^m}(l)$ over $\mathbb{P}^m$ for $l \geq 1$. Then we have ...
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52 views

Direct proof for $\mathcal{O}_{\mathbb{CP}_{1}}(-2) \cong T^{*}\mathbb{CP}_{1}$

Recall the holomporphic line boundle $\mathcal{O}_{\mathbb{CP}_{1}}(-2):= \mathcal{O}_{\mathbb{CP}_{1}}(-1) \otimes \mathcal{O}_{\mathbb{CP}_{1}}(-1)$ where $\mathcal{O}_{\mathbb{CP}_{1}}(-1)$ is the ...
3
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0answers
83 views

Polarization of abelian variety made by the sum of two divisors

Let $X$ be an abelian variety of dimension $n$, and let $L$ be a polarization, that is, an ample line bundle on $X$, with $\chi(L)=3$. In my specific case, I have that $L=\mathcal{O}_X(\Theta + D)$, ...
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36 views

Does the canonical line bundle coincide with dualizing sheaf when the base scheme is arbitrary

Let $X$ be a smooth projective curve over an algebarically closed field $k$. Consider the commutative diagram where the left square is cartesian. where $S,S'$ are schemes over $k$ and $X_S:=X\times ...
5
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1answer
147 views

Remembering Riemann-Roch

Embarrassingly, I've always struggled to remember the form of the Riemann-Roch theorem for curves. Does anyone have any intuition to share about how to remember the some of the terms in the formula? ...
5
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0answers
74 views

An application of Serre Duality

This is problem 2 of Chapter 5 in Voisin's book Hodge Theory and Complex Algebraic Geometry I. Let $X$ be a connected compact complex manifold of dimension $n$ and let $L$ be a holomorphic line ...
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1answer
82 views

Line Bundle with Everywhere Nonzero Section is Trivial

I'd like to show that a topological line bundle $(\pi,E,B)$ is trivial if there exists a section $\sigma : B \to E$ such that $\sigma(b) \neq 0$ for all $b$. I've been considering the map $B \times \...
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1answer
95 views

Canonical bundle and canonical divisor in a $K3$ surface

The algebraic definition of a $K3$ surface is this: A smooth algebraic suface $X$ is called $K3$ if: i) $X$ has trivial canonical bundle; ii) $h^1(X,\mathcal{O}_X)=0$. I know that i) means that the ...